Pion-Nucleus Scattering in the (3,3) Resonance Region

A many-body formulation of pion-nucleus scattering in the (3,3) resonance region is presented. It is shown that the self-energy (optical potential) of a pion propagating through a Fermi gas of nucleons can be expressed as an integral over density of an effective pion-nucleon forward-scattering amplitude $f(kq;kq)\mathrm{}$ in the medium. Using a pseudoscalar pion-nucleon interaction, one finds that $f(kq;kq)\mathrm{}$ satisfies a modified form of the standard Chew-Low equation for pion-nucleon scattering. The effect of the medium is twofold: (i) The effective pion-nucleon coupling constant is quenched in the medium, and (ii) the pion-nucleon threshold of the effective amplitude is moved upwards in energy. Both effects are due to Pauli-principle restrictions on nucleons in intermediate states. Like its free-space counterpart, $f(kq;kq)\mathrm{}$ displays a resonance behavior in the (3,3) pion-nucleon channel. The resonance appears as a characteristic energy dependence in the pion-nucleus optical potential calculated from $f(kq;kq)\mathrm{}$. The resonance position in pion-nucleus scattering is determined by three competing effects, and generally differs from the free pion-nucleon (3,3) resonance energy. One finds that: (i) The quenching effect moves the (3,3) resonance up in energy and narrows its width, (ii) the dispersive effect of the nuclear medium (increase of the real part of the wave number over the free-space value) moves the resonance downward, (iii) the energy dependence of the effective radius $R+\lambda$ seen by a pion of wavelength $\lambda$ also shifts the resonance downward in energy. The net energy shift is thus a result of a rather delicate cancellation of several competing effects. Absorption cross sections for negative pions are calculated for a range of nuclei. In general, the net downward energy shift with respect to the free (3,3) resonance is found to increase with increasing mass number.